Uniqueness and terminal attraction of upper transition attractors

Let $F_t$ be a one-parameter iterated function system satisfying properties (H1), (H2), and (H3) of Section 4. If $A_{\bullet}=A^{\bullet}$ for some upper transition attractor of $F_t$, then the upper transition attractor conjecture. $A^{\bullet}$ is the unique upper transition attractor of $F_t$ and is an attractor of $F_1$. This conjecture asks whether relationships known for a special type of one-parameter family extend to the more general setting described by (H1)–(H3).